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Binomial Distibutions. Target Goal: I can determine if the conditions for a binomial random variable are met. I can find the individual and cumulative binomial probabilities using the calculator. 6.3a h.w: pg 381: 61, 65, 66; pg 403: 69, 71, 73. The Binomial Distribution.

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binomial distibutions

Binomial Distibutions

Target Goal:

I can determine if the conditions for a binomial random variable are met.

I can find the individual and cumulative binomial probabilities using the calculator.

6.3a

h.w: pg 381: 61, 65, 66; pg 403: 69, 71, 73

the binomial distribution
The Binomial Distribution

When we are studying situations with two possible outcomes we are interested in a binomial setting.

What are some outcomes with two possible outcomes?

  • Toss a coin
  • Shoot a free throw
  • Have a boy or a girl
binomial setting
Binomial Setting

Suppose the random variable X = the number of successes in n observations.

Then X is a binomial random variable if:

  • Binary:there are only two outcomes, success or failure.
  • There is a fixed number n of trials.
  • The n trials are independent.
slide5
If X is a binomial random variable, it is said to have a binomial distribution, and is denoted as

B(n, p)

  • n: # of observations
  • p: probability of success
a binomial distribution or not
A binomial distribution or not?

Blood Types

Parents carry genes O and A blood type. The probability a child gets two “O” genes is 0.25. If there are 5 children in a family, is each birth independent?

  • Yes, so X is the count of success

(“O” blood type), and

X is B(5, 0.25)

slide7
Dealing cards
  • No. If you deal cards without replacement, the next card is affected by the previous. Not independent.
inspecting switches
Inspecting Switches
  • In a shipment, 10 % of the switches are bad (unknown to the inspector).
  • If the engineer takes a SRS of 10 switches from 10,000. The engineer counts X, the number of bad switches.
  • Is this Binomial?
slide9
Not quite binomial. Removing one changes the proportion of bad switches.
  • But, if the SRS is 10,000, removing one changes the remaining 9,999 very little.
  • When the population is much larger than the sample, we say the distribution is approximately binomial and very close to:
  • B(10,0.10)
activity a gaggle of girls
Activity: A Gaggle of Girls
  • How unusual is it for a family to have three girls if the probability of having a boy and a girl is equally likely?
  • If success = girl and failure = boy, then p(success) =

0.5.

slide11
Define the random variable X as the number of girls.
  • We want to simulate families with three children.
  • Our goal is to determine the long term relative frequency of a family with 3 girls, P(X=3)
using the random number table d
Using the Random Number Table D
  • Let the evendigitsrepresent “girl” and the odd digits represent “boy”.
  • Each student select their own row, and beginning at that row, read off numbers three at a time.
  • Each three digits will constitute one trail.
  • Use tally marks to record the results of 40 trails which will then be pooled with the class.
slide13
Calculate the relative frequency of the event.

P(X=3) = /40 =

  • vs. Class relative frequency:
using the calculator
Using the Calculator
  • Using the codes 1 = girl and 0 = boy;
  • Enter the command math:prb:randint(0,1,3).
  • This command instructs the calculator to randomly pick a whole number from the set {0,1} and do this three times.
  • The outcome {0,0,1} represents {boy, boy, girl}.
slide15
Continue to press ENTER until you have 40 trails.
  • Use a tally mark to record each time a {1,1,1} result.
slide16
Calculate the relative frequency of the event.

P(X=3) = /40 =

  • vs. class relative frequency:
  • Do the results of our simulation come close to the theoretical value for P(X=3) which is 0.125?
  • Even quicker…Try
slide17
Math:PRB:randBin(#trials,prob,#of simulations) randBin(3,.5,40) store L1
  • (2 1 0 3 …); 3 is 3 girls or next,
  • Sum(L1=3): count the # of 3 girl possibilities.
  • It changes the 3’s to 1(true) and counts.
finding binomial probabilities using the calculator
Finding Binomial Probabilities using the Calculator
  • The probability distribution function(p.d.f.) assigns a probability to each value of X.

P(X=2)

  • Calculator: TI-83:

2nd:VARS: binompdf (n, p, X)

cumulative distributions
Cumulative Distributions
  • The cumulative distribution function(c.d.f.) calculates the sum of the probabilities up to X.
  • P(X≤ 2) = P(X=0) + P(X=1) + P(X = 2)
  • Calculator: TI-83:

2nd:VARS: binomcdf (n, p, X)

example glex s free throws
Example: Glex’s Free Throws
  • Over an entire season Glex shoots 75% free throw percentage. She shoots 7/12 and the fans think she was nervous.
  • Is this unusual for Glex to shoot so poorly?
slide21
Studies of long series found no evidence that free throws are dependent so assume free throws are independent.
slide22
Probability of success = 0.75
  • X: The number of baskets made in 12 attempts.
  • Find the probability of making at most 7 free throws.

B(n, p)

B(12, 0.75)

slide23
P(X≤ 7) = P(X=0) + P(X=1) + …. + P(X = 7)
  • 2nd:VARS(dist):binomcdf (n, p, X)

binomcdf (12, 0.75, 7)

= 0.1576

  • Conclusion: Glex will make at most 7 out of 12 free throws about 16% of the time.
example type o blood type
Example : Type O Blood Type
  • Suppose each child born to John and Katie has probability 0.25of having blood type O. If John and Katie have 5 children, what is the probability that exactly 2 of them have type O blood?
  • Binomial distribution?
slide25
X: the number of children with type O blood.
  • Calculator: TI-83: binompdf(n, p, X)
  • Note: binompdf ( 5, 0.25, 0) = 0.2373, finds the P(X=0), none of the children have type O blood.
  • TI-89: tistat.binomPdf(n, p, X)
calculator procedure
Calculator Procedure:
  • Enter values of x; 0, 1, 2, 3, 4, 5 into L1
  • Enter the binomial probabilities into L2.
  • Highlight L2 and enter 2nd:VARS(dist):

binompdf (5, 0.25, L1).

  • Fill in table. (2 min)
plot a histogram of the binomial pdf
Plot a histogram of the binomial pdf.
  • Deselect or delete active functions in Y = window.
  • Define Plot1 to be a histogram with Xlist: L1, Freq: L2
  • Set the window X[0, 6]1 and Y[0, 1]0.1
use the trace button to inspect the heights of the bars
Use the TRACE button to inspect the heights of the bars.
  • Verify that the sum of the probabilities is 1.

STAT:CALC:1-VAR Stats L2

slide30
Fill in the following table of the cumulative distribution function (c.d.f.) for the binomial random variable, X.
  • To calculate cumulative probabilities:
  • Highlight L3 and enter

2nd:VARS(dist): binomcdf (5, 0.25,L1).

construct a histogram of the c d f
Construct a histogram of the c.d.f.:
  • Define Plot1 to be a histogram with Xlist: L1, Freq: L3
  • Use the window X[0, 6]1 and Y[0, 1]0.1.
  • Use the TRACE button to inspect the heights of the bars.
slide32
What do the heights represent?

The cumulative total at each X value.

compare
Compare:
  • p.d.f.
  • c.d.f.